Download Our (represented) World: A Quantum

Our (represented) World: A Quantum-Like Object
1
2,3
Ariane Lambert-Mogiliansky and François Dubois
1
Paris School of Economics,
48 Bd Jourdan, 75014 Paris, France.
2
Department of Mathematics, University Paris Sud,
Bât. 425, F-91405 Orsay Cedex, France.
3
Conservatoire National des Arts et Métiers, Paris, France,
Structural Mechanics and Coupled Systems Laboratory.
[email protected], [email protected].
March 2015
Abstract.
It has been suggested that observed cognitive limitations may be an
expression of the quantum-like structure of the mind. In this paper we explore some implications of this hypothesis for learning i.e., for the construction of a representation of the
world. For a quantum-like individual, there exists a multiplicity of mentally incompatible
(Bohr complementary) but equally valid and complete representations (mental pictures) of
the world. The process of learning i.e., of constructing a representation involves two kinds
of operations on the mental picture. The acquisition of new data which is modelled as a
preparation procedure and the processing of data which is modelled as an introspective
measurement operation.
This process is shown not converge to a single mental picture
but can oscillate forever. We dene a concept of entropy to capture relative intrinsic uncertainty. The analysis suggests a new perspective on learning. First, it implies that we
must turn to double objectication as in Quantum Mechanics: the cognitive process is the
primary object of learning. Second, it suggests that a representation of the world arises as
the result of creative interplay between the mind and the environment. There is a degree
of freedom that modies the objective of rational learning.
Keywords: learning, Bohr complementarity, information, uncertainty.
AMS classication: 81Q99, 91C99.
1
Ariane Lambert-Mogiliansky and François Dubois
Introduction
It is a common place that human beings are not capable of holding very complex picture in
mind (cf "small worlds" in Savage 1954). We consider reality focusing on one perspective at
a time and show diculties in combining perspectives as amply documented. This inability
to seize reality in its full richness suggests that the process of developing an understanding
of the world may not look like a puzzle that is assembled progressively. Instead, the human
mind may exhibit structural limitations in terms of the incompatibility of perspectives in a
way similar to properties in quantum mechanics. Ambiguous pictures of the kind provided
in diag. 1 provides a suggestive illustration of this phenomenon.
Quantum Mechanics is a theory about the physical world.
particles.
Its object is sub-atomic
The well-known "strange" features arise when an observer attempts to learn
about a sub-atomic system by means of a measurement device. A fundamental concept
in QM is Bohr complementarity: two properties can be incompatible in the sense that
they cannot have a determinate value simultaneously. As a consequence it is impossible to
experimentally access that physical world without interacting with it in a non-deterministic
manner. Intrinsic indeterminacy fundamentally changes the way we have to think about
the physical reality or at least about the physical reality that we can have some knowledge
about.
It introduce non-separability between the measurement process and the world.
There is no (accessible) given "outside world" but a contextual world in which we play an
active role.
For some time the quantum paradigm has been proposed to described among other
things aspects of the human psyche including preferences, attitudes and judgements in decision theory and psychology
)
(Lambert-Mogiliansky 2009, Busemeyer Bruza 2012, Khrenikov
knowledge
2010 among others . In this paper we take a step forward in recognizing that
is also an (psychic) object which in itself may exhibit non-classical features and we derive
some implications of this hypothesis for learning.
The approach can be viewed as a reversal of standard QM in the following sense. A
person considers a classical object e.g., a state agency, but her knowledge about it i.e., the
mental picture in her mind can behave in a non-classical way. Since human beings makes
decisions on the basis of perceived reality i.e., of mental pictures, our approach extends the
possible practical relevance of the quantum paradigm beyond sub-atomic physics to much
of macroscopic reality.
The most important element of our theory is Bohr-complementarity of mental perspectives of one and the same object. They parallel that of properties for sub-atomic particles
e.g., spin along dierent angles. We propose that Bohr complementarity of perspectives
captures the cognitive limitations that are responsible for our diculties to synthese information along dierent perspectives into one single coherent picture. The diculties arise
from the impossibility for incompatible perspectives to simultaneously have determinate
value. Just as in quantum physics, the system (here our mental picture) makes discrete
jumps when attempting (by means of introspection) to nd a determinate value along a
series of incompatible perspectives.
1
1
Let us take an example. we are interested in learning about a state agency. We may learn everything
about the recruitment system which informs about the qualications of people in various positions. It is a

Our (represented) World: A Quantum-Like Object
duckrabit.png
Figure 1.
What do you see ?
We model the cognitive process of learning as involving two essentially dierent operations. The rst operation corresponds to receiving information framed according to some
perspective.
We model this as a preparation procedure.
The second operation entails
processing information or updating the mental picture in terms of one's preferred perspective. It is modeled as an introspective measurement operation. This learning process is
not Bayesian as soon as we allow for Bohr complementarity of perspectives. The issue of
learning by non-classically minded agents has been approached in Danilov and Lambert
Mogiliansky [4, 5] and more recently in Busemeyer and Bruza [1].
This paper formalizes learning in the context of complete perspectives as a consequence
the analysis focuses on the starkest implications of the quantum paradigm including total loss of previously held "information". In the concluding section we discuss how this
investigation should be complemented by conveying for some role for memory.
The present paper builds on a model introduced in Dubois and Lambert-Mogiliansky
(2014).
It generalizes the basic model to allow for incomplete information in addition
to intrinsic uncertainty. Most importantly it shifts the focus to knowledge and learning
that we identify with the construction of a representation of reality. The approach unveils
the essentially creative nature of this process and its relative freedom with respect to
a "hypothetically objective" outside world.
It also invites us to rethink the concept of
information as a physical (psychic) entity not just an epistemic feature.
A central result in learning theory is that Bayesian updating (under weak conditions)
converges to the truth. However critics have been raised on several aspects of Bayesianism
(see e.g.
Gilboa, Postlewaite and Schmeidler [13].
Departures from Bayesian updating
may occur because they undervalue or overvalue priors (e.g., Epstein Noor Sandroni [10]).
Alternative approaches have been developed in connection with Maxmin Expected Utility (Gilboa and Schmeidler [12]) and Multiple priors (Hanany and Klibano[14]).
Our
personnel perspective that we can map back to a perspective guided by say a concern about the correctness
of a decision. If we next learn about the allocation of responsibility and associated path of decisions, it
activates other concepts (another network of neurones). We then try to map back into a perspective, we
nd out that it is not consistent with out earlier held beliefs.

Ariane Lambert-Mogiliansky and François Dubois
approach implies a novel departure from Bayesian updating appealing to the recent success of the quantum formalism in psychology and in social sciences (for a survey of recent
advances see Khrenikov [16] and Busemeyer and Bruza [1].
1
The basic model
The parallel with QM is as follows: the system is the represented world, it is a psychic
object. At any point in time the individual's knowledge about the (represented) world is
captured by the current mental picture or state. We denote it
|ϕi ∈ H
dimensional Hilbert space of mental pictures of the (relevant) world.
where
H
is nite
The state can be
characterized from dierent perspectives. A perspective is modelled as an operator similar
to an observable more precisely a complete set of commuting observables. It operates on
the mental picture to generate an outcome, a possible (set of ) picture(s) belonging to that
perspective. The present analysis addresses incomplete knowledge due to either incomplete
information or quantum indeterminacy or both.
1.1
The notion of measurement
Before entering into the analysis let us remind of some basic notions and how they relate
to our issue.
Generally, a measurement is an interaction between a system and some
measurement device, which yields some result, the outcome of the measurement that we
2 Two measurements are compatible if they, roughly speaking, can
can observe and record.
be performed simultaneously or more precisely, if the performance of one measurement
does not aect the result of the other. Suppose that the rst measurement gave outcome
o;
then we perform the second measurement and the rst one anew. In case we are dealing
with compatible measurements we obtain outcome
o
with certainty. If all measurements
are compatible we can substitute them with a single nest (complete) measurement, which
3 Performing that measurement we learn everything about the system.
is also rst-kind.
Such a system is classical.
The existence of incompatible measurements is a distinctive
feature of non-classical systems.
It is closely related to the impact of measurements on
the state and to the existence of dispersed states.
A state is dispersion-free when the
outcome of every possible measurement is uniquely determined, there is no reason for the
state to change. If all pure states are dispersion-free then measurements do not impact on
pure states and therefore all measurements are compatible. On the contrary, if a state is
dispersed then by necessity it will be modied by an appropriate measurement. On the
other hand, the change in a pure state is the reason for incompatibility of measurements.
The initial outcome of a rst measurement is not repeated because the system has been
modied by a second measurement (see Danilov and Lambert-Mogiliansky [4, 5]).
2
For instance in the case of the SternGerlach experimental setup, we let the electron travel through
a non-homogeneous magnetic eld and observe deviation either up or down. In another example we let
people play the Prisoner Dilemma (or the UG) and observe their choice. In the present context we see
part of the cognitive process (see below) as a measurement of the mental picture.
3
If we can perform a measurement twice in a row. If the outcomes of the two measurements always
coincide, we say that the measurement is a rst-kind measurement.

Our (represented) World: A Quantum-Like Object
In this paper a measurement is an introspective operation that acts on the agent's
mental picture.
The agent's asks himself a question: what are the implications of new
information for my understanding of a specic matter? and an answer, the outcome of the
mental process is brought to consciousness e.g., my understanding implies a recommendation for action. We propose that some cognitive limitations documented by practitioners
(give references) can be suitably modelled as the result of incompatible mental operations
that act on the mental picture to modify it.
1.2
Pure and mixed mental states
A pure state is a maximal information state. It may be the eigenstate of some perspective
so it gives maximal and precise knowledge of the world along some perspective.
Or it
may a superposed state that is not the eigenstate of any perspective (observable). Such
a state also contains maximal knowledge but is characterized by dispersion entropy (see
denition below).
No measurement can be performed on a pure state and provide new
4
information without losing some information.
In contrast a mixed state is a familiar
probabilistic combination of states which reects incomplete knowledge.
Measurements
can be performed on a mixed state to reach a maximal information pure state.
In this
section we briey discuss how a mixture of mental states can emerge.
At a rst a mixed state seems to be a most natural mental object. So for instance the
"world" is a friend's mood. You may be able to assign a probability to her willingness to
go out with you. Although that situation appears to be most ordinary, we need to make
clear how such a mixed state may arise in our context. In the world of QL-minded agents
there exist three possible scenario:
- the state has been prepared to a mixed state i.e., information provided in terms of a
5
probability distribution over possible states ;
- an introspective complete measurement is initiated but not completed. The initial
superposition has decohered into a probabilistic mixture of the possible states but not
collapsed into one of the eigenstates;
6
- a superposed state has spontaneously decohered;
In the present paper we shall not consider spontaneous decoherence as we do not explicitly consider time evolution. We focus on probabilistic mixture induced by the environment
through well-identied new information or by means of an introspective mental processes.
1.3
Intentionality: a preferred basis
We shall assume that an individual endeavours to construct a representation i.e., learn
about the world
because he has some concern in mind e.g.,
he has to select an action.
This assumption is consistent with psychological and neurobiological evidence that human
cognitive processes (including perception) are structured by some form of intentionality.
4
When the state already is a the eigenstate of the observable than no information is either lost or
gained.
5
You know your friend received the results of a determinant exam but only the probability for the
outcomes, if she passed she is in a good mood to go out.
6
You have reected over her mood and formulated the uncertainty without "solving it" for yourself.

Ariane Lambert-Mogiliansky and François Dubois
R that applies on
{r1 , ...rn } interpreted as
A perspective is dened as an "observable" that is an operator
state or mental picture. It is characterized by its eigenvalues
possible eigenpictures belonging perspective
R
the
the
. A perspective is a "complete set of com-
muting observables" so that a eigenstate in any perspective is a maximal information state.
R 6= P
Consequently and importantly any two perspectives
yield alternative descriptions
of the same "world". A pure mental picture can written as a superposition of the possible
eigenpictures in any perspective of the (relevant) world:
|ϕi =
X
λi |ri i,
X
i
λ2i = 1, λi ∈ R
j
which can also writes as a density operator
ρ = |ϕihϕ|
When the initial mental picture is
not
a pure state but a statistical mixture of pure states,
it can only be represented with a density operator
ρ=
X
τi |ϕi ihϕi | ,
ρ:
τi ≥ 0 ,
i
where
vector
X
τi = 1
j
|ϕi ihϕi | is a classical ket-bra notation to represent the orthogonal projector on the
|ϕi i. The density operator allows to express a situation where the agent's mental
state is characterized by both intrinsic uncertainty and incomplete information.
An action problem is dened as a correspondence from
of mental pictures of the world and
A
D : H → A where H is the space
the set of actions. For each (mentally represented)
state of the world, it denes which actions the agent wants to undertake.
Hypothesis
1
There exists a perspective
R∗
such that
D
is a coarsening of
R∗ .7
Hypothesis 1 implies that if the agent receives (maximal) information framed in perspective
D.
R∗
he will know exactly what to do. We also say that
Generally,
D
has a lower dimensionality than
H,
R∗
is fully congruent with
i.e., distinct eigenpictures can induce
the same action.
1.4
The learning process.
In our context, it is in place to distinguish between two essentially dierent operations in
8
the learning process . We shall present them as two successive phases.
Phase 1: Information acquisition
The rst phase of the mental process is an inter-
action between the mind (mental picture) and the outside world. The individual acquires
new information expressed within some perspective. This is a very important point: information is not "neutral", it is always framed i.e., it comes together with a perspective
7
A measurement M 0 is coarser than M if every eigenset of M is contained in some eigenset of M 0 , see
Danilov and Lambert-Mogiliansky [4, 5] p. 334.
8
Describing the two phases as essentially dierent is a simplication because information acquisition is
not "direct" either. It also involves mental processes: perception is not neutral and not always classical
(cf quantum zeno eect). Nevertheless a major distinction between the 2 phases is that the rst involves
an input from the outside while the second does not.

Our (represented) World: A Quantum-Like Object
(model) of the world. Because it is framed, to integrate a piece of information, the individual may have to (temporarily) switch to the perspective in which it is formulated. Either
the information belongs to the initial perspective
P 6= R.
R
or to some alternative perspective
In both cases information acquisition corresponds to a fully deterministic evolu-
tion of the mental picture. It operates on the initial state(picture) so as to project it onto
the one picture encoding the information content provided.
9
phase 1 resembles a process of preparation : ρ
In the terminology of QM,
→
ρ0 is the new density matrix.
P 0
0
If the information is consistent with R, ρ =
i θi |ri ihri |,
P
P
ρ = i θi |ri ihri |, i θi = 1.
0
where θi are calculated according
Let
ρ0 where
to the familiar Bayes rule. We have a classical case of
6= R, the
|pj i writes:
belief updating. The interesting case isn
when
oP
into the perspective
P
and
ρ0
in basis
ρ0 =
X
γi |pi ihpi | ,
X
10
mental picture is then "forced"
γi = 1,
i
Depending on the provided information content the density matrix may be a pure state or
mixed state (if the information content is probabilistic).
A rst central point to emphasize is that since
P 6= R
i.e., we are dealing with in-
compatible (alternative) perspectives, information embodied in
the process. A preparation in
P
ρ
the initial state is lost in
modies the mental picture into a specic mental picture
distinct from the initial one and the result is the same whatever the initial state in
R.
Again this would not have been the case if the provided information was in terms of the
same perspective as the initial state
R.
Of course this is a simplication but it it aims at
capturing the following. The loss of information reects the fact that the two pieces of informations (ρ and
ρ0 ) do not form a consistent picture for the individual.
It is as if they are
expressed in two dierent languages that do not translate into each other unambiguously
and the individual cannot but think (form a mental picture) in either one. In reality, the
process is of course much more complex. In particular, as earlier mentioned the individual
may keep some memory.
Phase 2: Updating
The second phase is a purely mental operation which does not
appeal to the outside world. It is an introspective measurement operation. It measures the
mental state
ρ0
with respect to the preferred perspective
R∗ .
This operation may be con-
ceptually decomposed into two steps in a way originally proposed by von Neuman (1932).
The rst step corresponds to the formulation of the mental picture in terms of a convex
combination (probabilistic mixture) of possible outcomes in the preferred perspective
R∗ .
This corresponds to making the measurement but not registering the result. The state is
projected on the preferred basis and decohered. The second step entails a decision - one
(or a subset of possible) result is singled out. It corresponds to registering the result of the
measurement. We shall refer to these two steps as respectively analysis and decision.
a. Analysis
9
In the process of preparation a system is put into a specic state.
Clearly the individual could neglect the information and he would not be forced into anything. But if
he wants to integrate it, he must exert some mental exibility and switch to the perspective in which that
information takes its meaning.
10

Ariane Lambert-Mogiliansky and François Dubois
Generally, the concerns that determine intentionality may not require a complete measurement. So both the analytical step and the decision step may be coarse (as opposed to
ne or complete measurement). In that case the outcome will entail the preservation of
some coherence. The analytical step and the state obtained after the decision is a superposition. We next describe the process for a complete measurement which fully decoheres
the state into a mixture of eigenpictures of
R∗
before selecting one of them. The state
∗
expressed in terms of the projectors associated with the eigenvectors of R
ρ0
is
:
!
ρ0 =
X X
i
k,l
The
ρ0
γi hrk∗ ||pi ihpi ||rl∗ i |rk∗ ihrl∗ |
contains o the diagonal elements which do not correspond to any eigenstate of
R∗ , they cannot be observed. The analysis step associated with a complete measurement
∗
corresponds to projecting the state into the R perspective, i.e., decoherence takes place
so only diagonal elements are left:
P ∗ ∗ 0 ∗ ∗
ρ0 → ρ∗ =
j |rj ihrj | ρ |rj ihrj | and
X
ρ∗ =
θk∗ |rk∗ ihrk∗ |
we write
ρ∗ :
k
∗
with diagonal coecients θj
=
P
i γi |
hpi | rj∗ i|2 ,
∗
j θj
P
= 1.
This new density operator is
the result of "analysis" in the sense that the possible results have been identied with
their respective probability for occurrence. Some information from the preparation stage
is "transmitted" modulo the correlation coecients
|hri∗ | pj i|
which are the probabilistic
∗
weights attached to the eigenstates in ρ . Some information is "lost" with the disappearance
of the non-diagonal coecients.
The correlation coecients linking two representations
are features of the mind i.e., of the relationship between alternative perspectives for the
individual; they may but need not reect features of the world that is represented. This is a
fundamental distinction between quantum cognition and quantum mechanics. In Quantum
Mechanics the correlation coecients pertains to observables and capture the interaction
between a physical object and the measurement instruments used to learn about it. The
subject matter of the present investigation is
not the world but the represented world which
is a mental construct. In particular, our model applies to the representation of classical
objects.
The coecient captures the quantum-like structure of the mind actualized in
cognitive activity.
b. Decision
The decision step is a non-deterministic process. When the updating phase corresponds
11 .
to a complete measurement, the resulting density operator represents an eigenstate
Decision selects one of the
|rj∗ ihrj∗ |
with the corresponding probability
θj∗
ρ∗ → ρ∗∗ = |rj∗ ihrj∗ |.
At this point, the individual has a maximally informed mental picture in
formation in
P
R∗
and all in-
0
from ρ has been "lost". Recall that it aected the mental process through
the probability for obtaining either one of the possible results. We see that after completing the cognitive process corresponding to interpreting the information in his own mental
11
The state resulting from a coarse measurement is a pure state that is a superposition of eigenstates.

Our (represented) World: A Quantum-Like Object
model, the agent's actual state of information is a maximal information state which cannot
be easily traced back to the initial
ρ
ρ0
or even to intermediary
and
ρ∗ .
We next develop a
simple numerical example.
1.4.1 A numerical example
We provide a simple numerical example illustrating the learning process dened above.
We consider a two dimensional Hilbert space and a 2X2 density matrix in the canonical
preferred basis
R∗ .
Let
ρ
represent the initial state
!
.25 0
.
0 .75
ρ=
(1.1)
ρ = .25|r1∗ ihr1∗ | + 0.75|r2∗ ihr2∗ |
Phase 1: Information acquisition
P with two eigenpictures {p1 , p2 } . The basis
{|r1∗ i, |r2∗ i } by an angle of π/8 . The agent
receives incomplete information in P i.e., the true state is either p1 or p2 with respec√
√
tive probability γ1 = (2 +
2)/4 and γ2 = (2 − 2)/4. The density operator repreConsider an alternative perspective
{|p1 i, |p2 i }
corresponds to a rotation of
senting the state that obtains as a result of the acquisition of the new information is
√
2)/4|p1 ihp1 | + (2 − 2)/4|p2 ihp2 |.
encapsulated in ρ has been lost.
ρ0 = (2 +
√
We readily sees that the initial information
Phase 2: Updating
The density operator must now be written in terms of the
.75|r1∗ ihr1∗ |
+
0.25|r2∗ ihr2∗ |
+
0.25|r1∗ ihr2∗ |
+
ρ0 =
(1.2)
ρ00 =
If we compare the matrices
ρ
and
ρ00 ,
projectors
ρ0 =
!
.75 .25
.
.25 .25
After the analysis step of the updating phase the matrix
(1.3)
|ri∗ ihri∗ |
.25|r2∗ ihr1∗ | or
ρ0
given in (1.2) becomes
!
.75 0
.
0 .25
we observe that the probabilities associated with
the eigenpictures of the preferred perspective have been inverted so if we complete the
measurement, there is now 75% chance that the agent ends up being convinced of
r1
instead of 25% as before he acquired new information. In order to fully understand why
this is a reection of cognitive constraints we need to introduce the concept of dispersion
entropy and explore further the example which we do below.
2
2.1
Learning under irreducible uncertainty
Dispersion entropy - a measure of doubt
In this section we propose a concept of entropy that captures the amount of (relative)
uncertainty that characterizes the mental picture of an agent. The rst important point

Ariane Lambert-Mogiliansky and François Dubois
is that this uncertainty is not absolute but
relative
to a specic perspective e.g., corre-
sponding to some objective that requires action. To make the point most clear we shall
therefore conne ourselves to pure states i.e., states of maximal information.
In those
states, the corresponding mental picture is a superposition of eigenpictures i.e., the only
uncertainty that is left is due to the quantum-like structure of the mind.
Shannon entropy which measures uncertainty associated with a classical probability is
dened for a discrete probability distribution {pj } according to
S=−
X
pj log pj .
j
It has been generalized by von Neumann to apply to quantum states and in particular to
density operators. Let
ρ=
P
λi |ϕi i hϕi |:
S (ρ) = −tr (ρ log ρ)
or equivalently for
S (ρ) = −
X
λi log λi .
As does Shannon entropy, von Neuman entropy measures the amount of uncertainty
to incomplete information
while we are interested in measuring
therefore talk about the entropy of
dispersion
intrinsic
due
uncertainty. We
12
rather than of probability distribution.
The relative entropy of dispersion is dened:
Denition The dispersion entropy of a pure state ρ = |ψi hψ| with |ψi =
1 relative to a specic perspective A with eigenvectors {|ai i} is
P
λi |ϕi i ,
P
λ2i =
. X
S (ρ, A) = −
αj log αj
where αj =
2
i λi | haj |
P
ϕi i|2 ,
P
αi = 1.
A few important remarks:
1. Dispersion entropy is a relative concept, it varies with the perspective;
2. Dispersion entropy is equivalent with the von Neuman entropy of the mental picture
ρ0
=
P
αi |ai i
decohered by the measurement of
A
but before the decision (see analysis in
Phase 2 of section 2.2 ).
3.
S (ρ, A) = 0
when the mental picture
|ϕi
is an eigenpicture of
A
and is positive
otherwise.
In contrast with the standard concept of entropy we cannot say that the dispersion entropy of a pure state measures how much information we might
gain by actually performing
the measurement - because we are in a zero von Neuman entropy state. The
pure state
ρ
S (ρ, A) of
A on
captures the non epistemic impact of the operation of measurement of
the mental picture .
13
12
See sect 2.2 above.
Clearly the denition could be extended to include arbitrary state i.e., mixed state. It would then
capture the combined uncertainty from incomplete information and intrinsic uncertainty and would remain
a relative concept.
13

Our (represented) World: A Quantum-Like Object
In the context of quantum cognition we would like to link dispersion entropy to a
14
psychological state of "doubt".
S (ρ, A)
is then a measure of the extent of "inner un-
15 An introspective mea-
certainty". It can be resolved by an introspective measurement.
surement does not only take the logical consequence of information (the projection that
16 Another
decoheres the state), it also includes a resolution (what we called "decision").
way to put it is that a (complete) introspective measurement operates on a pure mental
picture to resolve a state of doubt or lack of conviction.
We next develop a simple argument showing how additional information can increase
dispersion entropy relative to the preferred basis. For the purpose of the argument, we assume that the agent's mental state is initially an eigenpicture of his preferred perspective|ϕi
|r2∗ i. This state reects information that is perfectly congruent with R∗ but is framed in
P
R : |ϕi =
λi |ri i (= |r2∗ i) . The agent is not aware of r2∗ unless he updates "his beliefs"
∗
which corresponds to the (measurement) operation of R . If he did, he would obtain
R∗ |ϕi = |r2∗ i
He would then be fully determined i.e., convinced with respect to the decision problem.
There would not be any (relevant) uncertainty left with
γ2 = 1 we have S (ρ, R∗ ) = 0.
But
assume that instead he acquires new information relative to another but related aspects of
P. This corresponds to steps 1 of the cognitive process
0
prepared into |ϕ i equal to some |pi i i.e., information is
the issue. We denote this perspective
|ϕi is
P. "Enriched"
above. The mental state
maximal in perspective
with this new information, the agent now updates
his mental picture with respect to his preferred perspective (step 2):
|ϕ0 i = |p2 i =
X
γj |rj∗ i →
∗
any|rj i with probability
γj2 6= 1
j
We see that while he previously potentially knew
r2∗
(but was unaware of it) he is now
is a state of hesitation where he believes that there is only a probability of
γ22 < 1
that
∗ 17 The acquisition of new information has triggered the loss of
the true eigenpicture is r2 .
"conviction" reecting previously held information which is in eect lost leaving the agent
in a state of intrinsic uncertainty captured by a strictly positive dispersion entropy:
X
S ρ0 , R ∗ = −
γj2 log γj2 > 0.
j
where
ρ0 =
P
j
γj2 |rj∗ ihrj∗ |.
Proposition Assume that the agent's mental state is one eigenpicture of his preferred
perspective, acquiring additional information in terms of a perspective that is not compatible
with the preferred one aects the mental state so as to increase dispersion entropy reecting
doubt or lost conviction.
14
The term doubt is used in it s psychological meaning i.e., not the cartesian rational doubt that denes
the scientic approach.
15
It could also be resolved by information acquisition in perspective A.
16
Another way to put it is that an introspective measurement is a way to deal with "doubts" where
doubt is not (only) a problem of information but an inner state of lack of conviction.
17
After the introspective process, he will end up believing r2∗ with some probability less than 1.

=
Ariane Lambert-Mogiliansky and François Dubois
This is an important result. From the point of view of "inner conviction" additional
information is unambiguously detrimental when the starting point is an eigenpicture of the
preferred perspective as illustrated in the argument above: the new information necessarily
induces uncertainty with respect to the agent's action relevant perspective. More generally
i.e., starting from an arbitrary eigenpicture (possibly mixed and with non zero relative
dispersion entropy) as in the example above, new information may either decrease or
increase dispersion entropy. In the next section we discuss possible implications for rational
learning.
Remark 1. In our case r2∗ is true because it reects an information congruent with R∗ and
|pi i is also true. The agent is not mistaken. He simply cannot hold in one single picture
both informations. The introspective operation when processing information p2 involves the
whole (cognitively limited) mind and therefore upsets earlier held beliefs..
We illustrate the result in Proposition 1 in gure 2 where the dimensionality of a
perspective is 2. The broad line corresponds to the preparation stage and the thin lines to
the measurement.
fig2.jpg
Figure 2.
Detrimental information
Example cont.
Using the numerical example above, we illustrate the impact of new information on
dispersion entropy. The starkest results appears when starting out with
ρ = |r2∗ ihr2∗ |
and
assume that the state was generated as the result of the acquisition of maximal information
∗
√ R . After
2+ 2
0
acquiring information the state of knowledge/conviction is ρ = |p1 ihp1 | =
|r1∗ ihr1∗ | +
4
√
√
√
√
2
|r1∗ ihr2∗ | + |r2∗ ihr1∗ | + 2−4 2 |r2∗ ihr2∗ |. Then ρ00 = 2+4 2 |r1∗ ihr1∗ | + 2−4 2 |r2∗ ihr2∗ |. The new
4
√
√ 2+ 2
2+ 2
0
∗
state is associated with the following dispersion entropy S(ρ , R ) = −
log
−
4
4
√ √
2− 2
2− 2
∗
= 0.4165 > S(ρ, R ) = 0.
4 log
4
in perspective
R∗
and thus associated with zero dispersion entropy relative to
So now we readily see that new information created a state of doubt when there should
not have been any.
ciated with
|p1 ihp1 |
The doubt does not arise as the result of the information content assobut exclusively as the result on the cognitive constraint expressed by
the incompatibility between the two perspectives corresponding to

R∗
and
P
respectively.
Our (represented) World: A Quantum-Like Object
Remark 2. It must be emphasized that the reasoning above relies on our assumption
that we are dealing with a classical counter part where the agent learns that r∗ = r2∗ with
probability 1 (corresponding to the pure state in the quantum model). If we relax slightly this
assumption (i.e., allowing for incomplete information), the result in proposition 1 could be
approached with a classical model of a dierent avor. Assume the agent has priors that
puts nearly all the weight (but not all) on r2∗ so α1 6= 0 and α3 6= 0 as in b in ??. Next,
he receives information p2 . If according to the agent's structural model, that information is
highly correlated with r1∗ (which is not the case in the original model) he updates his beliefs
and puts more weight on ω1 and ω3 than he initially had. He is now less certain about what
to do than he was before he got the information. His new beliefs exhibit a distribution with
higher entropy. However, he has not lost information as in the quantum case. Instead the
interpretation is that his priors were wrong so new information brought him closer to the
truth.
In view of the remark above a legitimate question is: are we only talking about interpretation or do quantum cognitive limitations really have a more profound bearing on
learning? The next section addresses this question.
2.2
Rational Learning under Irreducible Uncertainty
Learning whether it is by classically minded agent or QL-minded ones is always about
the construction of a mental representation of the world. In our context it is important
to distinguish between two properties of a representation.
On the one hand the mental
picture may be more or less informationally complete. And on the other hand, the mental
picture may be more or less connected with the outside world it is supposed to represent.
In a classical context this distinction also exists and information from the outside world is
used to complement and test the mental representation. If the picture is incomplete, new
information will bring it closer to the outside truth.
If the model is wrong attempts to
update it with new information will give rise to inconsistencies. It should be stressed that
no satisfactory approach exists as to how a classically minded agent proceeds in face of
data revealing failures of the underlying model. There is no obvious incremental path of
modications (the notion of distance is ambiguous at best) and certainly no prescription
for discrete jumps between alternative models.
For a quantum-like agent, the situation is similar except that there is not one single "true" model (mental picture) but a multiplicity of Bohr complementary (correlated)
"true" models. Discrete jumps between representations are parts of the cognitive process
by force of complementarity. With quantum cognition the link between the epistemic state
(the mental picture) of the represented world and the world in itself is ambiguous. In fact
the term "epistemic state" does not seem proper in face of intrinsic indeterminacy as the
multiplicity of zero entropy states suggests.
As repeatedly emphasized by cognitive sci-
entists we cannot address knowledge without simultaneously addressing human cognitive
activity. As we shall see this takes a precise meaning in our context.

Ariane Lambert-Mogiliansky and François Dubois
2.2.1 Oscillating forever and disconnect.
Bayesian learning operates within a Boolean algebra. The objective is to learn the parameters of the model of the system.
18 Provided the priors are not inconsistent with the true
model, we know (Schwartz 1965) that starting from any such priors Bayesian updating
converges to the true state.
With quantum learning, we are in a Hilbert space, that is there exists a variety of
resolutions
of the system
i.e.,
a variety of perspectives, of valid theories of the system.
Therefore, the state of knowledge of a quantum minded individual does not converge with
new information but can oscillate forever. Consider again the example above. After the
agent performed the introspective measurement
his understanding in
P,
R∗ ,
if the agent is being asked about
he will perform a new measurement of the eigenpicture resulting
from the rst measurement and
ρ0
will not be recovered. Performing those measurements
alternately, he will keep oscillating without converging i.e., without being able to settle for
a denite value in both
P
and
R∗
simultaneously.
and the agent might simply remember
ρ0 .
19 Of course this example is simplistic
But in more sophisticated context, we expect
the modication of the mental picture to be eective.
In Quantum Mechanics, irreducible uncertainty gave rise to a rich philosophical debate about the nature of reality. (see e.g., D'Espaganat 1979). In our context when the
irreducible uncertainty is a consequence of the quantum structure of the mind, new questions arise.
The mental picture easily "drift away" from a "truthful representation" of
the outside world.
20 We next briey discuss what rational learning may mean in such a
situation.
2.2.2 The object of QL-learning and "Knowledge about the world"
In quantum mechanics when physicists were confronted with the impossibility of rst hand
objectication, they resorted to second hand objectication. That is basically the Hilbert
space model of quantum mechanics. It captures how measurements impact on the system
and how measurements are related among each other. The state vector being the central
object that encapsulates all information and it is the basis for making predictions. Similarly, the situation we are facing in quantum cognition suggests that the primary object of
scientic learning must be the human mind i.e., the structure of mental perspectives, how
they aect the mental representation of the world and how they are related to each other.
The current mental picture is the state of the representation, it encapsulate the information
known to the individuals and it is the basis for making prediction and further operations.
18
Some consider also Bayesain updating with multiple priors (see e.g., Hanany and Klibanov [14]). But
there is no consensus as to how to proceed - in sharp contrast with Bayesian updating of single priors.
19
The general result is a transposition into cognition of the basic feature of quantum mechanics namely
that it is not possible for complementary properties to have a determinate value simultaneously.
20
The role of the acquisition of new information from an incompatible perspective is neither a complement
nore a test of the truthfulness of the current picture. But it is an operation that forces a (temporary)
reconnection between the mental picture and the outside world. Only information within the perspective
of the mental picture is a true test in a way similar to learning in the classical world with classically minded
agents.

Our (represented) World: A Quantum-Like Object
21
This structure is assumed to exhibit invariance reecting psychological regularities.
Hence we propose that the primary object of QL-learning is the human cognitive process
- the non-boolean algebra of mental perspectives.
What knowledge about the outside
world concern, we have seen that for QL-minded agents knowledge about the world is a
mental picture that reects both inputs from the outside world (information) and input
from the inside world (decision in introspective measurement). At any point of time the
representation we have of the world depends on the path of information acquisition and
introspective measurements performed. By choosing a specic path, the individual chooses
let be in a non-deterministic manner, her "world". She is the creator of the world she lives
in. We know from psychologists that people systematically underscore their responsibility
in failure (relative to bad luck) and overscore their impact on success. It has been suggested
that this can be fully rational to optimize motivation and self-image.
22
The quantum
cognitive approach provides a framework for eectively acting in this manner, it internalize
the (emotional) constraints that justify lying to oneself into cognition itself - see the section
on Sources of indeterminacy. Since QL-agents do have some freedom in the determination
of the world they live in - it only seems rational to create a world in which we are more
productive and feel good.
Rational learning about the world transforms into rational choices that aects our
well-being in a way similar to other choices.
not everything is possible.
As all creative processes, it is constrained
Moreover one should not end up with a mental picture that
is fully disconnected with the outside world because that undermines the value of our
(other) actions which are based on that representation. But we are invited to recognize a
signicant freedom. We thus propose that rational construction of a representation of the
world entails selecting what information to acquire and which introspective measurements
to perform so as to produce a mental picture that maximizes our well-being in the actual
world.
For instance when having very specic practical concerns adequacy between the
mental picture and the outside world may be particularly valuable. In such a case a strong
conviction obtained after a series of introspective measurements may benet from being
tested by new
information. But one should be careful about the congruence between perspectives. A
piece of information from a highly non-congruent perspective may unsettle conviction in
a non-productive way. Future research is called upon to characterize rational learning by
QL-agent in terms of optimal decision-making with respect to information acquisition and
introspective measurements.
Concluding remarks
This paper is a rst exploration of learning by cognitively limited agents where the limitations are modeled appealing to quantum like characteristics of the mind. A main motivation
is that the quantum approach has shown successful in explaining behavioral anomalies in
decision-making while it also seems able to capture concerns expressed by practitioners
21
At least we do expect some invariance at the individual level.
The idea is that motivation is not fully maleable and must be incited: if you feel bad about your self
you e.g., do not nd the energy to make bold moves.
22

Ariane Lambert-Mogiliansky and François Dubois
dealing with information transmission and communication. In particular, the fact that i.
people reason about reality within the frame of some perspective (mental script) or as we
call it, a mental picture, ii, the multiplicity of potential representations of reality generates
specic problems for learning. We model cognitive limitations in terms of the multiplicity
of Bohr complementary mental representations of one and the same reality. We nd that
under intrinsic uncertainty, additional information may induce doubts where conviction
was in adequacy with previously held information.
Our analysis calls for a new approach to learning. rst, it puts the cognitive process
at the center: that is the logical structure behind the geometry of perspectives. Second,
it unveils a form of creative freedom in the construction of a mental representation that
invites a fully dierent approach.
Namely learning about the world becomes a part of
decision-making whose objective is to maximize well-being.
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